Kontsevich quantization formula

In mathematics, the Kontsevich quantization formula describes how to construct a generalized ★-product operator algebra from a given arbitrary finite-dimensional Poisson manifold. This operator algebra amounts to the deformation quantization of the corresponding Poisson algebra. It is due to Maxim Kontsevich.

Deformation quantization of a Poisson algebra
Given a Poisson algebra $(A, {⋅, ⋅})$, a deformation quantization is an associative unital product $$\star$$ on the algebra of formal power series in $ħ, A ħ$, subject to the following two axioms,


 * $$\begin{align}

f\star g &=fg+\mathcal{O}(\hbar)\\ {}[f,g] &=f\star g-g\star f=i\hbar\{f,g\}+\mathcal{O}(\hbar^2) \end{align}$$

If one were given a Poisson manifold $(M, {⋅, ⋅})$, one could ask, in addition, that


 * $$f\star g=fg+\sum_{k=1}^\infty \hbar^kB_k(f\otimes g),$$

where the $B_{k}$ are linear bidifferential operators of degree at most $k$.

Two deformations are said to be equivalent iff they are related by a gauge transformation of the type,
 * $$\begin{cases}

D: A\hbar\to A\hbar \\ \sum_{k=0}^\infty \hbar^k f_k \mapsto \sum_{k=0}^\infty \hbar^k f_k +\sum_{n\ge1, k\ge0} D_n(f_k)\hbar^{n+k} \end{cases}$$ where $D_{n}$ are differential operators of order at most $n$. The corresponding induced $$\star$$-product, $$\star'$$, is then
 * $$f\,{\star}'\,g = D \left ( \left (D^{-1}f \right )\star \left (D^{-1}g \right ) \right ).$$

For the archetypal example, one may well consider Groenewold's original "Moyal–Weyl" $\star$-product.

Kontsevich graphs
A Kontsevich graph is a simple directed graph without loops on 2 external vertices, labeled f and g; and $n$ internal vertices, labeled $Π$. From each internal vertex originate two edges. All (equivalence classes of) graphs with $n$ internal vertices are accumulated in the set $G_{n}(2)$.

An example on two internal vertices is the following graph,
 * [[Image:KontsevichGraph2.svg|Kontsevich graph for n=2]]

Associated bidifferential operator
Associated to each graph $Γ$, there is a bidifferential operator $B_{Γ}( f, g)$ defined as follows. For each edge there is a partial derivative on the symbol of the target vertex. It is contracted with the corresponding index from the source symbol. The term for the graph $Γ$ is the product of all its symbols together with their partial derivatives. Here f and g stand for smooth functions on the manifold, and $Π$ is the Poisson bivector of the Poisson manifold.

The term for the example graph is


 * $$\Pi^{i_2j_2}\partial_{i_2}\Pi^{i_1j_1}\partial_{i_1}f\,\partial_{j_1}\partial_{j_2}g.$$

Associated weight
For adding up these bidifferential operators there are the weights $w_{Γ}$ of the graph $Γ$. First of all, to each graph there is a multiplicity $m(Γ)$ which counts how many equivalent configurations there are for one graph. The rule is that the sum of the multiplicities for all graphs with $n$ internal vertices is $(n(n + 1))^{n}$. The sample graph above has the multiplicity $m(Γ) = 8$. For this, it is helpful to enumerate the internal vertices from 1 to $n$.

In order to compute the weight we have to integrate products of the angle in the upper half-plane, H, as follows. The upper half-plane is $H ⊂ $\mathbb{C}$$, endowed with the Poincaré metric


 * $$ds^2=\frac{dx^2+dy^2}{y^2};$$

and, for two points $z, w ∈ H$ with $z ≠ w$, we measure the angle $φ$ between the geodesic from $z$ to $i∞$ and from $z$ to $w$ counterclockwise. This is


 * $$\phi(z,w)=\frac{1}{2i}\log\frac{(z-w)(z-\bar{w})}{(\bar{z}-w)(\bar{z}-\bar{w})}.$$

The integration domain is Cn(H) the space
 * $$C_n(H):=\{(u_1,\dots,u_n)\in H^n: u_i\ne u_j\forall i\ne j\}.$$

The formula amounts
 * $$w_\Gamma:= \frac{m(\Gamma)}{(2\pi)^{2n}n!}\int_{C_n(H)} \bigwedge_{j=1}^n\mathrm{d}\phi(u_j,u_{t1(j)})\wedge\mathrm{d}\phi(u_j,u_{t2(j)})$$,

where t1(j) and t2(j) are the first and second target vertex of the internal vertex $j$. The vertices f and g are at the fixed positions 0 and 1 in $H$.

The formula
Given the above three definitions, the Kontsevich formula for a star product is now
 * $$f\star g = fg+\sum_{n=1}^\infty\left(\frac{i\hbar}{2}\right)^n \sum_{\Gamma \in G_n(2)} w_\Gamma B_\Gamma(f\otimes g).$$

Explicit formula up to second order
Enforcing associativity of the $$\star$$-product, it is straightforward to check directly that the Kontsevich formula must reduce, to second order in $ħ$, to just


 * $$\begin{align}

f\star g &= fg +\tfrac{i\hbar}{2}\Pi^{ij}\partial_i f\,\partial_j g -\tfrac{\hbar^2}{8}\Pi^{i_1j_1}\Pi^{i_2j_2}\partial_{i_1}\,\partial_{i_2}f \partial_{j_1}\,\partial_{j_2}g\\ & - \tfrac{\hbar^2}{12}\Pi^{i_1j_1}\partial_{j_1}\Pi^{i_2j_2}(\partial_{i_1}\partial_{i_2}f \,\partial_{j_2}g -\partial_{i_2}f\,\partial_{i_1}\partial_{j_2}g) +\mathcal{O}(\hbar^3) \end{align}$$